How to calculate the parameters of the local coordinate system. Local coordinate systems

In the historical past, they were tied to the area of ​​research in which topographic surveys were carried out to compile maps. To carry out this work, it was necessary to select an initial reference point and orient it relative to some characteristic direction, for example, north according to the compass. Or these could be directions to distant points with expected long-term safety. And relative to this initial direction, which can be considered the beginning of the coordinate system, fix all objects on the surface. In different regions and countries, different orientation systems were chosen, and all the results of the work differed from each other.

On the European territory of the country, by 1932, the leveling of the state network, which began under the program of its construction in 1928, was completed. The SK-32 coordinate system appeared, which was developed in Western Siberia, on the territory of Kazakhstan and received the name “Pulkovskaya”.

In the Far Eastern region and East Siberian regions, separate geodetic networks have developed since 1934, again one can say in the local coordinate system. These include the Svobodnenskaya, Aldanskaya, Khabarovsk systems. By connecting the Pulkovo and Svobodnenskaya reference systems in 1936 through geodetic points in the Krasnoyarsk region, actual planned residuals with the following values ​​were obtained:

  • Δx= -270m;
  • Δy= +790m.

During the development of state geodetic justification in the Central Asian region, the Tashkent local system was used, on the Kamchatka peninsula - the Petropavlovsk one, in the north-eastern district - the Magadan local coordinate system. Absolute altitude coordinates also came from various level surfaces, nearby seas from the Baltic to the Japanese, as well as the Black, Caspian and Okhotsk.

Transition from MSC to the general state system and back

Deviations in the coordinates of points in the Pulkovo and Svobodnensky systems by almost 800 meters, even over significant distances of 7000 km, led to certain assumptions. The conclusions about the discrepancy between the accepted parameters of the Bessel ellipsoid, determined in 1841, and the actual dimensions of the Earth, were later confirmed. According to new calculations of Krasovsky’s reference ellipsoid, the discrepancy in the values ​​of the semimajor axis with the Bessel ellipsoid was 845 m. After leveling all included points of the astronomical and geodetic network from Pulkovo to the Far East, a unified state coordinate system of 1942 (SK-42) was created.

On the basis of the national SK-42 in 1963, a new coordinate system (SK-63) distributed throughout the country was created. At this time, a qualitative technological space leap was taking place, after the launches of the first artificial earth satellites. Presumably SK-63 arose with special distortions relative to SK-42 in different regions according to different parameters for the purpose of additional secrecy. Although the nature of secrecy has hardly changed with the advent of the new SK-63. Perhaps even information data has become more accessible, including to surveyors and cartographers. The algorithm for its construction, of course, was completely secret. The 1963 coordinate system was built in blocks, covering the entire country. That is, almost every block could be considered a local reference frame.

The most amazing thing was what happened next with SK-63. Initially, its appearance was considered as the emergence of a new state coordinate system, but at the same time on the basis of geodetic points SK-42 with all its errors. Since it was created using recalculation from SK-42 with angular turns and linear displacements along the coordinate grid for each zone, at the edges of each of which overlaps are possible. Thus, it can also be characterized as a set of local coordinate systems interconnected.

In addition, it should be noted that SK-63 is still not a Gauss-Kruger projection and methods for reducing and determining corrections for this component are not specified in it. But the most interesting thing is that they always seemed to strive for the largest set, coverage of points and polygons in geodetic networks, their equalization, to determine the parameters of the Earth, the ability to determine coordinates at any point on the earth’s surface and establish unified coordinate systems. In SK-63, the opposite happened. Probably, first of all, because of the secrecy regime. On the other hand, perhaps its original purpose was to be used for territories limited to an area of ​​five thousand square kilometers within different subjects of the state.

This leads to a definite conclusion that a local coordinate system can be considered any conventional reference system within a limited area with mandatory reference to the coordinate grid through transition parameters, the so-called “keys”. If this requirement is not met, such a system is considered conditional. Very often, conditional systems are used for small single construction projects in the city. MSC is intended for conducting topographical work, land surveying, and maintaining cadastral records in the regions.

The current situation with MSC

The practical situation with the geodetic economy of the country since the times of the USSR and the development of regions has led to the situation of using SK-63 as a local system for drawing up large-scale maps and master plans in cities in the departments of architecture and land management. In 1988, it seems that its use was canceled. But due to the presence of large archival funds of topographic plans and due to the fact that nothing was proposed to replace the SK-63, it has been used for a long time. Already in another country, all this heritage led to the idea of ​​​​creating local coordinate systems for each region. This has been implemented in practice since the 2000s.

In Russia, the initiative to establish local coordinate systems belongs to the executive bodies of the federal and regional authorities. Since 2007, programs have been developed and adopted in each region to introduce automation of the land cadastre, real estate register and regulations on MSC.

The approved provision is considered the main document establishing the MSC. It, as a rule, contains all the necessary information for transforming coordinate systems using certain algorithms laid down in GOST R 51794-2001 “Coordinate systems” by transformation methods for transition:

  • from the general earth SC WGS-84 (PZ-90) to MSC and back;
  • from the national SK-95 again to MSC and also back.

The “Regulations on MSK”, as a rule, indicate:

  • nomenclature numbers of all sheets of maps at a scale of 1:100000 on which the MSK is formed;
  • the total area of ​​the area it covers;
  • transformation parameters, the so-called “transition keys”, from the state geocentric (PZ-90) to local SC. They include seven quantities: shifts along the X, Y, Z axes (Δx, Δу, Δz), rotation angles around the X, Y, Z axes (Wx, Wy, Wz), scale factor.
  • parameters for the transition from the national SK-95 to the local SK, also in the amount of seven parameters;
  • root mean square errors of transformation of plan coordinates and UCS of elevation marks;
  • parameters of the mathematical surface in the MSC, which is taken as the Krasovsky ellipsoid with compression values ​​equal to 1/298.3 and semi-major axis 6378245m;
  • Gaussian projection parameters for calculating flat coordinates in the MCS. These include shifts of the MSK along the abscissa (X), ordinate (Y) axes, the scale factor on the adopted axial meridian and the longitude value of the axial meridian;
  • lists of all geodetic points, respectively, in a flat MCS with rectangular coordinates and a spatial MCS with geodetic coordinates.

MSCs in the regions are installed to carry out all geodetic work on their territories:

  • topographical;
  • exploration;
  • construction;
  • land management;
  • cadastral;
  • when operating unique structures;
  • other special works.

GOST R 51794-2008

Group E50

NATIONAL STANDARD OF THE RUSSIAN FEDERATION

Global navigation satellite systems

COORDINATE SYSTEMS

Methods for transforming coordinates of defined points

Global navigation satellite system and global positioning system. Coordinate systems. Methods of transformations for determinate points coordinate

OKS 07.040
OKSTU 6801

Date of introduction 2009-09-01

Preface

The goals and principles of standardization in the Russian Federation are established by Federal Law of December 27, 2002 N 184-FZ "On Technical Regulation", and the rules for applying national standards of the Russian Federation are GOST R 1.0-2004 "Standardization in the Russian Federation. Basic provisions"

Standard information

1 DEVELOPED 29 by the Research Institute of the Ministry of Defense of the Russian Federation

2 INTRODUCED by the Technical Committee for Standardization TC 363 "Radio Navigation"

3 APPROVED AND ENTERED INTO EFFECT by Order of the Federal Agency for Technical Regulation and Metrology dated December 18, 2008 N 609-st

4 INSTEAD GOST R 51794-2001


Information about changes to this standard is published in the annually published information index "National Standards", and the text of changes and amendments is published in the monthly published information index "National Standards". In case of revision (replacement) or cancellation of this standard, the corresponding notice will be published in the monthly published information index "National Standards". Relevant information, notifications and texts are also posted in the public information system - on the official website of the Federal Agency for Technical Regulation and Metrology on the Internet

Amendments have been made, published in IUS No. 4, 2011, IUS No. 6, 2011, IUS No. 9, 2013

Amendments made by the database manufacturer

1 Application area

1 Application area

This standard applies to coordinate systems that are part of the geodetic parameter systems "Earth Parameters", "World Geodetic System" and the coordinate base of the Russian Federation, and establishes methods for transforming coordinates and their increments from one system to another, as well as the procedure for using numerical values ​​of elements transformation of coordinate systems when performing geodetic, navigation, cartographic work using the equipment of consumers of global navigation satellite systems.

2 Terms and definitions

The following terms with corresponding definitions are used in this standard:

2.1 semimajor axis of the ellipsoid : A parameter characterizing the size of the ellipsoid.

2.2 reference ellipsoid: An ellipsoid adopted for processing geodetic measurements and establishing a geodetic coordinate system.

2.3 geodetic coordinate system: A system of parameters, two of which (geodetic latitude and geodetic longitude) characterize the direction of the normal to the surface of the reference ellipsoid at a given point in space relative to the planes of its equator and prime meridian, and the third (geodesic height) represents the height of the point above the surface of the reference ellipsoid.

2.4 geodetic latitude: The angle between the normal to the surface of the reference ellipsoid passing through a given point and the plane of its equator.

2.5 geodetic longitude: The dihedral angle between the planes of the geodesic meridian of a given point and the prime geodesic meridian.

2.6 geodetic height: The height of a point above the surface of the reference ellipsoid.

2.7 geodetic meridian plane: A plane passing through the normal to the surface of the reference ellipsoid at a given point and parallel to its minor axis.

2.8 plane of the astronomical meridian: A plane passing through a plumb line at a given point and parallel to the Earth's axis of rotation.

2.9 plane of the prime meridian: The plane of the meridian from which longitudes are calculated.

2.10 geoid: An equipotential surface that coincides with the surface of the World Ocean in a state of complete rest and equilibrium and continues under the continents.

2.11 equipotential surface: A surface on which the potential has the same value.

2.12 Global Positioning System(Global Positioning System): Global navigation satellite system developed in the USA.

2.13 Earth's gravitational field; GPZ: Gravity field on the Earth's surface and in outer space, caused by the Earth's gravitational force and the centrifugal force resulting from the Earth's daily rotation.

2.14 quasigeoid: A mathematical surface that is close to the geoid and serves as a reference surface for establishing a system of normal heights.

2.15 space geodetic network; KGS: A network of geodetic points that fix a geocentric coordinate system, the position of which on the earth's surface is determined from observations of artificial Earth satellites.

2.16 World geodetic system(World Geodetic System): A system of geodetic parameters developed in the United States.

2.17 model of the Earth's gravitational field: Mathematical description of the characteristics of the Earth's gravitational field.

2.18 normal height: The height of a point above the quasi-geoid, determined by the geometric leveling method.

2.19 normal gravitational field of the Earth: The Earth's gravitational field, represented by the normal gravity potential.

2.20 terrestrial ellipsoid; OSE: An ellipsoid whose surface is closest to the geoid as a whole, used to process geodetic measurements over the entire surface of the Earth in a common terrestrial (geocentric) coordinate system.

2.21 planetary model of the Earth's gravitational field: A model of the Earth's gravitational field, reflecting the gravitational characteristics of the Earth as a whole.

2.22 ellipsoid compression : A parameter characterizing the shape of the ellipsoid.

2.23 system of geodetic parameters of the Earth: A set of numerical parameters and accuracy characteristics of the fundamental geodetic constants of the earth's ellipsoid, the planetary model of the Earth's gravitational field, the geocentric coordinate system and the parameters of its connection with other coordinate systems.

2.24 fundamental geodesic constants: Mutually consistent geodetic constants that uniquely determine the shape of the general Earth ellipsoid and the normal gravitational field of the Earth.

2.25 elements of transformation of coordinate systems: Parameters used to convert coordinates from one coordinate system to another.

2.26 flat rectangular coordinates: Plane coordinates on the plane on which the surface of the reference ellipsoid is displayed according to a certain mathematical law.

3 Abbreviations and symbols

The following abbreviations and symbols are used in this standard:

3.1 GLONASS is a global navigation satellite system developed in the Russian Federation.

3.2 GPS is a global navigation satellite system developed in the USA.

3.3 GGS - state geodetic network.

3.4 GPZ - Earth's gravitational field.

3.5 KNS - space navigation system.

3.6 WGS; The World Geodetic System is a system of geodetic parameters developed in the USA.

3.7 OZE - common terrestrial ellipsoid.

3.8, , , - axes of the spatial rectangular coordinate system.

3.9 PZ; Earth Parameters is a system of geodetic parameters developed in the Russian Federation.

3.10 SC - coordinate system.

3.11 - semimajor axis of the general earth ellipsoid in the PZ system.

3.12 - semimajor axis of the global ellipsoid in the WGS system.

3.13 - semimajor axis of the Krasovsky ellipsoid.

3.14 - compression of the general earth ellipsoid in the PZ system.

3.15 - compression of the global ellipsoid in the WGS system.

3.16 - compression of the Krasovsky ellipsoid.

4 Systems of geodetic parameters

4.1 System of geodetic parameters "Earth Parameters"

The PP system includes: fundamental geodetic constants, OZE parameters, the PP coordinate system fixed by the coordinates of points of the space geodetic network, characteristics of the GPZ model and transformation elements between the PP system and the national reference coordinate systems of Russia. The numerical values ​​of the transformation elements between the PP system and the national reference coordinate systems of Russia and the order of their use when transforming coordinate systems are given in Appendices A, B.

Notes

1 for use for geodetic support of orbital flights and solving navigation problems, the geocentric coordinate system “Earth Parameters 1990” (PZ-90) was given the status of a state coordinate system.

2 By Order of the Government of the Russian Federation dated June 20, 2007 N 797-r, in order to improve the tactical and technical characteristics of the global navigation satellite system GLONASS, improve geodetic support for orbital flights and solve navigation problems, an updated version of the state geocentric coordinate system "Earth Parameters 1990" was adopted for use " (PZ-90.02).

3 The numerical values ​​of transformation elements between the PZ-90.02 and PZ-90 coordinate systems and the order of their use when transforming coordinate systems are given in Appendix D.


The theoretical definition of the PZ coordinate system is based on the following provisions:

a) the origin of the coordinate system is located at the center of mass of the Earth;

b) the axis is directed to the International Conditional Origin;

c) the axis lies in the plane of the prime astronomical meridian established by the International Time Bureau;

d) the axis complements the system to the right coordinate system.

The positions of points in the PP system can be obtained in the form of spatial rectangular or geodetic coordinates.



The center of the OSE coincides with the origin of the PZ coordinate system, the axis of rotation of the ellipsoid coincides with the axis, and the plane of the prime meridian coincides with the plane.

Note - The reference surface in the systems of geodetic parameters PZ-90 and PZ-90.02 is taken to be a common earth ellipsoid with a semi-major axis of 6378136 m and a compression of 1/298.25784.

4.2 System of geodetic parameters "World geodetic system"

The WGS parameter system includes: fundamental geodetic constants, a WGS coordinate system fixed by the coordinates of points of the space geodetic network, OSE parameters, characteristics of the GPZ model, transformation elements between the WGS geocentric coordinate system and various national coordinate systems.

The numerical values ​​of transformation elements between the PZ coordinate system and the WGS coordinate system, as well as the order of using transformation elements, are given in Appendices C and D.

Note - On January 1, 1987, the first version of the WGS-84 coordinate system was introduced. On January 2, 1994, a second version of the WGS-84 coordinate system was introduced, designated WGS-84(G730). On January 1, 1997, the third version of the WGS-84 coordinate system was introduced, designated WGS-84(G873). The fourth version of the WGS-84 coordinate system is currently in effect, designated WGS-84(G1150) and introduced on January 20, 2002. In the following designations for versions of the WGS-84 coordinate system, the letter "G" means "GPS", and "730", "873" and "1150" indicate the GPS week number corresponding to the date to which these versions of the WGS-84 coordinate system are assigned .

The theoretical definition of the WGS coordinate system is based on the provisions given in 4.1.

WGS point positions can be obtained as spatial rectangular or geodetic coordinates.

Geodetic coordinates refer to the OZE, the dimensions and shape of which are determined by the values ​​of the semi-major axis and compression.

The center of the ellipsoid coincides with the origin of the WGS coordinate system, the axis of rotation of the ellipsoid coincides with the axis, and the plane of the prime meridian coincides with the plane.

Note - The reference surface in WGS is a global ellipsoid with a semi-major axis of 6378137 m and a compression of 1/298.257223563.

4.3 Reference coordinate systems of the Russian Federation

The coordinate base of the Russian Federation is represented by a reference coordinate system, implemented in the form of the GGS, which fixes the coordinate system on the territory of the country, and the state leveling network, which extends the system of normal heights (the Baltic system) to the entire territory of the country, the initial origin of which is the zero of the Kronstadt footpole.

The positions of the defined points relative to the coordinate base can be obtained in the form of spatial rectangular or geodetic coordinates, or in the form of flat rectangular coordinates and heights.

Geodetic coordinates in the reference coordinate system of the Russian Federation refer to the Krasovsky ellipsoid, the dimensions and shape of which are determined by the values ​​of the semi-major axis and compression.

The center of the Krasovsky ellipsoid coincides with the origin of the reference coordinate system, the axis of rotation of the ellipsoid is parallel to the axis of rotation of the Earth, and the plane of the prime meridian determines the position of the origin of longitude calculation.

Notes

1 In 1946, a unified reference coordinate system of 1942 (SK-42) was adopted for the entire territory of the USSR. The reference surface in SK-42 is Krasovsky's ellipsoid with a semimajor axis of 6378245 m and a compression of 1/298.3.

2 By Decree of the Government of the Russian Federation of July 28, 2000 N 568, a new reference system of geodetic coordinates of 1995 (SK-95) was adopted for use in geodetic and cartographic work. Krasovsky's ellipsoid is taken as the reference surface in SK-95.

5 Methods for transforming the coordinates of defined points

5.1 Converting geodetic coordinates to rectangular spatial coordinates and vice versa

The transformation of geodetic coordinates into rectangular spatial coordinates is carried out according to the formulas:

where , , are the rectangular spatial coordinates of the point;

, - geodetic latitude and longitude of the point, respectively, rad;

- geodetic height of the point, m;

- radius of curvature of the first vertical, m;

- eccentricity of the ellipsoid.

The values ​​of the radius of curvature of the first vertical and the square of the eccentricity of the ellipsoid are calculated, respectively, using the formulas:

where is the semimajor axis of the ellipsoid, m;

- compression of the ellipsoid.

To convert spatial rectangular coordinates into geodetic coordinates, iterations are required when calculating geodetic latitude.

To do this, use the following algorithm:

a) calculate the auxiliary quantity using the formula

b) analyze the value:

1) if 0, then

2) if 0, at

c) analyze the meaning:

1) if 0, then

2) in all other cases, calculations are performed as follows:

- find auxiliary quantities , , using the formulas:

Implement an iterative process using auxiliary quantities and:

If the value determined by formula (16) is less than the established tolerance value, then

, (17)

; (18)

If the value is equal to or greater than the specified tolerance value, then

and the calculations are repeated starting from formula (14).

When transforming coordinates, the value (10) is taken as the tolerance for terminating the iterative process. In this case, the error in calculating the geodetic height does not exceed 0.003 m.

5.2 Transformation of spatial rectangular coordinates

Users of GLONASS and GPS systems need to perform coordinate transformations from the PP system to the WGS system and vice versa, as well as from PP and WGS to the reference coordinate system of the Russian Federation. These coordinate transformations are performed using seven transformation elements, the accuracy of which determines the accuracy of the transformations.

Elements of transformation between the PZ and WGS coordinate systems are given in Appendices C, D.

The conversion of coordinates from the WGS system to the coordinates of the reference system of the Russian Federation is carried out by sequentially transforming the coordinates first into the PZ system, and then into the coordinates of the reference system.

The transformation of spatial rectangular coordinates is performed according to the formula

where, , - linear elements of transformation of coordinate systems when moving from system A to system B, m;

, , - angular elements of transformation of coordinate systems when moving from system A to system B, rad;

- a large-scale element of transformation of coordinate systems when moving from system A to system B.

The inverse transformation of rectangular coordinates is performed according to the formula

5.3 Conversion of geodetic coordinates

The transformation of geodetic coordinates from system A to system B is performed according to the formulas:

where , - geodetic latitude and longitude, expressed in units of plane angle;

- geodetic height, m;

, , - corrections to the geodetic coordinates of the point.

Amendments to geodetic coordinates are determined using the following formulas:

where , - corrections to geodetic latitude, longitude, ...;

- correction to geodetic height, m;

, - geodetic latitude and longitude, rad;

- geodetic height, m;

, , - linear elements of transformation of coordinate systems during the transition from system A to system B, m;

, , - angular elements of transformation of coordinate systems when moving from system A to system B, ...;

- a large-scale element of transformation of coordinate systems when moving from system A to system B;

Radius of curvature of the meridian section;
- radius of curvature of the first vertical;

Major semi-axes of ellipsoids in coordinate systems B and A, respectively;

, - squares of eccentricities of ellipsoids in coordinate systems B and A, respectively;

- number of arc seconds in 1 radian [(206264.806)].

When transforming geodetic coordinates from system A to system B, the values ​​of geodetic coordinates in system A are used in formula (22), and when converting back - in system B, and the sign of the corrections , , in formula (22) is reversed.

Formulas (23) provide the calculation of corrections to geodetic coordinates with an error not exceeding 0.3 m (in a linear measure), and to achieve an error of no more than 0.001 m, a second iteration is performed, i.e. take into account the values ​​of corrections to geodetic coordinates using formulas (22) and repeat the calculations using formulas (23).

At the same time

Formulas (22), (23) and the accuracy characteristics of transformations using these formulas are valid up to latitudes of 89°.

5.4 Converting geodetic coordinates to plane rectangular coordinates and vice versa

To obtain flat rectangular coordinates in the Gauss-Kruger projection adopted on the territory of the Russian Federation, geodetic coordinates on the Krasovsky ellipsoid are used.

Flat rectangular coordinates with an error of no more than 0.001 m are calculated using the formulas

where , - flat rectangular coordinates (abscissa and ordinate) of the determined point in the Gauss-Kruger projection, m;

- geodetic latitude of the determined point, rad;

- the distance from the determined point to the axial meridian of the zone, expressed in radian measure and calculated by the formula

Geodetic longitude of the determined point, ...°;

The integer part of the expression enclosed in square brackets.

The transformation of flat rectangular coordinates in the Gauss-Kruger projection on the Krasovsky ellipsoid into geodetic coordinates is carried out according to the formulas

where , - geodetic latitude and longitude of the determined point, rad;

- geodetic latitude of a point, the abscissa of which is equal to the abscissa of the point being determined, and the ordinate is equal to zero, rad;

- number of the six-degree zone in the Gauss-Kruger projection, calculated by the formula

The integer part of the expression enclosed in square brackets;

- ordinate of the determined point in the Gauss-Kruger projection, m.

The values ​​of , and are calculated using the following formulas:

where is an auxiliary quantity calculated by the formula

Auxiliary quantity calculated by the formula

Abscissa and ordinate of the determined point in the Gauss-Kruger projection, m.

Error of coordinate transformation according to formulas (25); (26) and (32)-(36) is no more than 0.001 m.

5.5 Converting increments of spatial rectangular coordinates from system to system

The transformation of increments of spatial rectangular coordinates from coordinate system A to system B is carried out according to the formula

The inverse transformation of increments of spatial rectangular coordinates from system B to system A is performed according to the formula

In formulas (37) and (38), the angular transformation elements , , are expressed in radians.

5.6 Relationship between geodetic and normal heights

Geodetic and normal heights are related by the relation:

where is the geodetic height of the determined point, m;

- normal height of the determined point, m;

- height of the quasigeoid above the ellipsoid at the determined point, m.

The heights of the quasi-geoid above the reference ellipsoid of the geodetic parameter systems PP and WGS are calculated using GPZ models, which are an integral part of the geodetic parameter systems.

When recalculating the heights of a quasigeoid from coordinate system A to coordinate system B, use the formula

where is the height of the quasigeoid above the OSE, m;

- height of the quasigeoid above the Krasovsky ellipsoid, m;

- correction to geodetic height, calculated using formula (23), m.

Appendix A (mandatory). Elements of transformation between the refined coordinate system of the Earth Parameters and the reference coordinate systems of the Russian Federation

Appendix A
(required)

Conversion of coordinates from the 1942 reference coordinate system to the PZ-90.02 system

23.93 m; 0;
-141.03 m; -0.35;
-79.98 m; -0.79;
-130.97 m; 0.00;
-81.74 m; -0.13;
(-0.22)·10;

Conversion of coordinates from the PZ-90.02 coordinate system to the 1995 reference coordinate system

Appendix B (mandatory). Elements of transformation between the Earth Parameters coordinate system and the reference coordinate systems of the Russian Federation

Appendix B
(required)

Converting coordinates from the 1942 reference coordinate system to the PZ-90 system

25 m; 0;
-141 m; -0.35;
-80 m; -0.66;
0;

Converting coordinates from the PZ-90 coordinate system to the 1942 reference coordinate system

Converting coordinates from the 1995 reference coordinate system to the PZ-90 system

25.90 m;
-130.94 m;
-81.76 m;

Conversion of coordinates from the PZ-90 coordinate system to the 1995 reference coordinate system

Appendix B (mandatory). Elements of transformation between the refined coordinate system of the Earth Parameters and the coordinate system of the World Geodetic System

Appendix B
(required)

Converting coordinates from the PZ-90.02 coordinate system to the WGS-84 system

0.36 m; 0;
+0.08 m; 0;
+0.18 m; 0;
0;

Converting coordinates from the WGS-84 coordinate system to the PZ-90.02 system

Appendix D (mandatory). Elements of transformation between the Earth Parameters coordinate system and the World Geodetic System coordinate system

Appendix D
(required)

Converting coordinates from the PZ-90 coordinate system to the WGS-84 system

1.10 m; 0;
-0.30 m; 0;
-0.90 m; -0.20±0.01;
(-0.12)·10;

Converting coordinates from the WGS-84 coordinate system to the PZ-90 system

Appendix D (mandatory). Elements of transformation between the refined coordinate system PZ-90.02 and the coordinate system PZ-90

Appendix D
(required)

Converting coordinates from the PZ-90.02 coordinate system to the PZ-90 system

1.07 m; 0;
+0.03 m; 0;
-0.02 m; +0.13;
(+0.22) ·10;

Converting coordinates from the PZ-90 coordinate system to the PZ-90.02 system

Electronic document text
prepared by Kodeks JSC and verified against:
official publication
M.: Standartinform, 2009

To move from one coordinate system to another, there are fundamentally 2 types of transformations:

- coordinate transformation using officially published transformation parameters, also called global transformation methods, since they specify the algorithm for transition between coordinate systems as a whole, throughout the entire space of action of these coordinate systems, for example, between WGS-84 and SK-95, ITRF and SK-95, PZ-90 and WGS-84, etc.;

- transformation of coordinates using transformation parameters calculated using a limited set of control points located on the local territory, the coordinates of which are known in both of these CS, also called local conversion methods, since they specify a coordinate recalculation algorithm that operates only in relation to the local territory on which the control points are located.

Classic three-dimensional coordinate transformation methods used primarily for global transformations between spatial three-dimensional rectangular or ellipsoidal (geodesic) coordinate systems are the Helmert method and the Molodensky method, respectively.

Conversion from one spatial (3D) rectangular coordinate system X,Y,Z(SK-1) to another spatial system of rectangular coordinates (SK-2) according Helmert consists of performing three operations:

Transferring the beginning of CK1 to the beginning of CK2 by shifting along the axes XYZ by magnitudes T X, T Y, T Z, corresponding to the difference in coordinates of the origins of coordinate systems 1 and 2 (or, similarly, by the value of the coordinate values ​​of the final coordinate system SK-2 in the original SK-1);

Rotate around each of the coordinate axes by amounts w X , w Y , w Z ,;

Scaling (introducing a multiplier m, characterizing the change in the scale of the final SC-2 relative to the scale of the initial SC-1).

Thus, the Helmert transformation is specified by the 7 above parameters, which is why it is often called the 7-parameter transformation, or the Euclidean similarity transformation, and the transformation parameters included in it are called the Helmert parameters.



For the 7-parameter Helmert transform, the formula is used

Where [ X, Y, Z]SK1- coordinates of the point in the original coordinate system;

Where, [ X, Y, Z]SK2- coordinates of the point in the final coordinate system;

T X, T Y, T Z- the displacement value of the origin of coordinate system 1 along the corresponding axes to the origin of coordinate system CK2;

w X , w Y , w Z- rotation around each of the axes of the coordinate system;

m- scale factor taking into account the different scales of these SCs, its value is usually<10 -6 и дается в единицах 6-го знака после запятой.

Method Molodensky used to convert between two spatial geodetic coordinate systems B, L, H(i.e., eliminating the need to change to rectangular XYZ coordinates).

To transform coordinates using the Molodensky method, use the formula

. (5)

,

.

Classical 3D method Sometimes they carry out calculations of the 7-parameter transformation in two modifications: Bursa-Wolf and Molodensky-Badekas.

The difference between the modifications is that in the Bursch-Wolf transformation, the center of rotation is the origin of the original coordinate system A and the 7 above-described parameters of the Helmert transformation are used - CLASSIC.

and in the Molodensky-Badekas modification, the center of rotation is the “center of gravity” (a point on the work site that has average coordinates) of the control points in the original coordinate system A, therefore, in this modification of the classical three-dimensional transformation, 3 more parameters are added to the 7 Helmert parameters (coordinates of the center of rotation X 0 , Y 0 , Z 0). In LGO it is implemented like this

The scheme of coordinate transformations when performing geodetic work using GNSS technologies is given below

12. Free adjustment, varieties of minimally limited adjustment, limited adjustment, limited adjustment with simultaneous estimation of transformation parameters.

The procedure for mathematical processing of satellite measurements:

Ø processing of GNSS measurements and calculation of baselines,

Ø calculation of residuals of closed figures,

Ø assessment of measurement accuracy based on figure residuals,

Ø network equalization,

Ø accuracy assessment based on adjustment results

Tools for mathematical processing of satellite measurements– special commercial software for processing satellite measurements

Equalization Concepts

In general, the development of GNS through GNSS measurements involves determining the coordinates of a large number of stations with a limited number of GNSS receivers. The observations carried out in the project are divided into sessions consisting of observations at individual stations (points). The following methods for adjusting satellite observations have been developed and used:

· adjustment of observations made on one station (for the case of absolute (point) positioning);

· processing one baseline and subsequent integration of the baselines into a network,

· combined adjustment of all received observations of a single session ( adjustment of observations of many stations of one session), And

· combining solutions many sessions into a rigorous, all-encompassing network solution,

· combining satellite and traditional geodetic measurements.

Equalization one station(point positioning, “one-point” solution) provides absolute station coordinates in the WGS-84 (or PZ-90) system. If only code measurements are processed, then due to low accuracy these results are usually of little interest for geodetic applications, but they often meet the requirements of some geophysics, GIS and remote sensing applications. A typical area of ​​this application is navigation.

Concept single baseline very widely used in satellite data processing software. In joint adjustment, observations from two simultaneously operating receivers are processed, primarily in the form of double differences. The result is the components of the baseline vector and the corresponding covariance matrix K XYZ

Individual baselines are used as input data in network equalization program. Processing of observations in the network breaks down into primary adjustment(baseline solution) and secondary adjustment(equalization of baseline vectors).

Most manufacturers offer programs with their receivers that use the baseline concept. These programs are useful for small projects, field data verification, and real-time applications.

IN adjustment of many stations of one session all data that was observed simultaneously by all receivers participating in the session is jointly processed. In this case, the results of the solution are R-1 independent vectors and a covariance matrix of size 3( R- 1)´ 3( R- 1). Depending on the software available, results can also be given in sets of 3 R coordinates and a covariance matrix of size 3 R´3 R. The covariance matrix is ​​also block-diagonal, in which the size of the non-zero diagonal blocks is a function of the number of receivers R. Therefore, it is a strict adjustment of observations using all reciprocal stochastic relations. For geodetic purposes, such a “multipoint” adjustment has conceptual advantages over the baseline method, since the full potential of SRNS accuracy is used.

Multiple session solutions can be combined into equalization of many sessions or, more precisely, in solution for many stations and many sessions. This is a common technique when large networks are broken up due to a limited number of receivers. The main condition in such an adjustment is that each session is connected to at least one other session through one or more common stations at which observations were made in both sessions. Increasing the number of common stations increases the stability and reliability of the entire network.

Combining satellite and traditional types of measurements is necessary for the transition from global coordinates of satellite network points to the state reference system SK-95 and to the Baltic system of normal heights.

Leveling geodetic networks built using satellite technologies is a necessary stage in geodetic technology. The objectives of equalization are:

· coordination of the totality of all measurements in the network,

· minimization and filtering of random measurement errors,

· identification and rejection of rough measurements, elimination of systematic errors,

· obtaining a set of adjusted coordinates and the corresponding elements of the baselines with an assessment of accuracy in the form of errors or covariance matrices,

· transforming coordinates into the required coordinate system,

· converting geodetic heights to normal heights above a quasi-geoid.

Thus, the main goal of adjustment is to increase accuracy and present the results in the required coordinate system with an accuracy assessment.

There are free, minimally limited and limited (non-free) adjustment.

IN free adjustment All points of the network are considered unknown, and the position of the network relative to the geocenter is known with the same accuracy as the coordinates of the starting point of the network. In this case, the coefficient matrix of the system of correction equations (plan matrix) and, therefore, the normal matrix will have a rank defect equal to three. However, the use of the matrix pseudo-inversion apparatus used in some programs allows for adjustment. Its results reflect the internal accuracy of a network that is not distorted by errors in the original data.

When fixing the coordinates of one point, we get minimally limited adjustment, in which the normal matrix turns out to be non-degenerate. To achieve meaningful control, a vector network should not contain vectors whose ends are not connected to at least two stations.

Free and minimally constrained adjustment are used to solve the first three adjustment problems (harmonizing the totality of all measurements in the network, minimizing and filtering random measurement errors, identifying and rejecting rough measurements, eliminating systematic measurement errors).

When fixing more than three coordinates - limited adjustment. In this case, additional restrictions will be imposed in relation to the minimum necessary.

Limited adjustment is performed after successful completion of minimally limited adjustment to include the newly constructed network in the existing network, in its coordinate system, including the elevation system. To do this, the new network must be connected to at least two stations of the existing network.

A special problem is the joint adjustment of satellite and conventional geodetic measurements. Its essence is that traditional geodetic measurements (angle measurements, leveling, astronomical determinations, etc.) are performed using a level, that is, the geoid is used as a reference surface. Baseline measurements are made in the system of axes of a common earth ellipsoid. To correctly reduce data to one particular system, it is necessary to know the heights of the geoid above the ellipsoid with appropriate accuracy.

In case of limited adjustment, the following can be inserted as additional unknowns into the parametric equations: connection parameters between coordinate and elevation systems.

The combination of satellite and traditional measurements is carried out with limited adjustment. Mathematical models for spatial coordinates are based on the Helmert method (local transformation using the method of similarity of coordinates in Cartesian). In this transformation, the scale factor is the same in all directions, as a result of which the shape of the network is preserved, i.e. The angles are not distorted, but the lengths of the lines and the positions of the points may change.

For discussion.

One of the components of errors in satellite networks is the error in transforming field data from a geocentric CS (WGS-84), in which measurements are performed, into a reference CS (SK-95, SK-42, SK-63, MSK...), where the final coordinates of points are calculated networks.
The official communication parameters WGS-84 and SK-42, specified in GOST R 51794-2008, apply to the Pulkovo region (the beginning of SK-42). As it moves away, shift errors accumulate in SK-42, which in the regions of Siberia and the Far East can reach several meters. That is, local parameters in different regions may differ significantly from the officially known ones.
To determine (calculate) local communication parameters, coordinates of 4-5 points are needed, known in two systems. And if some coordinates (SK-42, SK-63, MSK...) can be obtained officially, then the exact coordinates of points based on WGS-84, as a rule, are not known. They are usually obtained from satellite measurements, where the network is calculated from one point, the coordinates of which in WGS-84 are obtained as navigation ones (autonomously, using on-board satellite ephemeris). The error in determining such coordinates (shift along X, Y) can be 2-3 meters or more. If the same points are observed at another time, or another group of points is taken in the same area, then different coordinate values ​​in WGS-84 will be obtained.
Consequently, it will not be possible to obtain accurate coordinates in WGS-84 and, accordingly, accurate communication parameters in this way. And the smaller the distance between localization “calibration” points, the more roughly the communication parameters between systems are determined.
However, ultimately, what is important to us is not the accuracy of determining the coordinates of points in WGS-84, but how much will errors in determining parameters affect the accuracy of converting vectors from WGS-84 to SK-42 (and other SCs based on the Krasovsky ellipsoid)?
Is it so important to determine local communication parameters every time? For example, working in the European part of Russia, where the distance from Pulkovo is not so great, where SK-42 has not yet been subjected to large distortions and these distortions are comparable to the errors in autonomous determination of coordinates in WGS-84? After all, it will not be possible to obtain more accurate parameters from autonomous coordinates (with an error of several meters).
Isn't it better to recalculate the coordinates of the initial points in WGS-84 using GOST parameters and use them for primary processing of satellite measurements?
Or immediately, using GOST parameters, configure the program to work in SK-42 (SK-63, MSK...)? It’s up to who it’s more convenient for and who works in what software.

Once upon a time, when starting my satellite measurements, I performed localization every time. Over time, several dozen points were accumulated, which were combined into a single network and refined communication parameters were obtained for a large number of points and over a large area. Comparing the vector increments converted from WGS to MSC according to refined and local parameters, I was convinced that there was no significant difference. Due to the reversal, the magnitude of the increments may differ slightly, but the length of the vector projection onto the MCS plane remains practically unchanged. The same thing happened when comparing the increments of vectors obtained using the refined and GOST parameters.
And this is in places where the local errors of the SK-42 reached 10 meters.
The error in calculating vector increments is several times smaller than the error in the relative positions of GGS points.
After adjustment to GGS points, the residuals of the increments are scattered, and the final coordinates of the determined points in both versions differ in the first millimeters.

I don’t want to say at all that it is always and everywhere that GOST parameters of communication between ICs need to be applied. This is probably not acceptable for long vectors or for handling cool networks. But in topographic work, when the starting points are not enough to determine local parameters, it is quite possible to use GOST ones. A network with sufficient control can rely on just 2-3 starting points.

Anyone can perform the experiment without going into the field. On your completed project, where the communication parameters between WGS-84 and SK-42 were previously determined by localization, replace the local parameters with GOST ones and re-process the measurements (before processing, do not forget to edit the coordinates of the starting points - they may change after replacing the communication parameters).
Compare the coordinates of the identified points from the two options and announce the resulting discrepancies “in the studio.” It would be interesting.